Sharp Minimum Research Articles

On Some Works of Boris Teodorovich Polyak on the Convergence of Gradient Methods and Their Development

The paper presents a review of the current state of subgradient and accelerated convex optimization methods, including the cases with the presence of noise and access to various information about the objective function (function value, gradient, stochastic gradient, higher derivatives). For nonconvex problems, the Polyak–Lojasiewicz condition is considered and a review of the main results is given. The behavior of numerical methods in the presence of a sharp minimum is considered. The aim of this review is to show the influence of the works of B.T. Polyak (1935–2023) on gradient optimization methods and their surroundings on the modern development of numerical optimization methods.

Influence of Potential and the Presence of Boric Acid on Pt1-xCox Alloy Electrodeposition and Magnetic Properties

The effects of potential and the presence of B(OH)3 on Pt1-xCox alloy electrodeposition from aqueous chloride-based solutions on Ru substrates was investigated. Films deposited at potentials more reducing than −0.65 V vs SCE were hexagonal close packed and greater than 90% Co (mole basis), and films deposited at potentials more oxidizing than −0.65 V were face centered cubic and showed a monotonic decrease in cobalt content as the potential increased. The composition and structure-potential dependences were not strongly affected by the presence of B(OH)3. Structural change coincided with a distinct knee-like feature in the composition-potential relationship, along with a prominent narrow voltammetric peak associated with Co deposition, possibly related to nucleation and growth of the hcp phase. The presence of B(OH)3 produced a sharp minimum in both Coulombic efficiency and deposition rate at potentials near −0.65 V and almost entirely suppressed the voltammetric feature. This may be associated with a combination of B(OH)3-derived proton reduction and inhibition of metal deposition by adsorbed B(OH)3 or B(OH)3-derived species. The presence of B(OH)3 affected the magnetic behavior of films deposited at potentials more oxidizing than −0.55 V (i.e., those with compositions less than about 40% Co) only weakly, but resulted in generally smaller maximum magnetizations for films deposited at more reducing potentials, and notably a much lower magnetization for films deposited at −0.65 V.

Dynamics of electron capture in positron‐hydrogen scattering under dense semi‐classical plasmas

AbstractThe scattering dynamics of electron capture in ‐H(1 s) scattering under dense semi‐classical plasma (DSCP) environments has been investigated theoretically. Coupled multi‐channel two‐body Lippmann‐Schwinger equations have been solved by retaining +H(1 s) and p + Ps(1 s) channels to calculate the cross sections (CS) of the electron capture process at intermediate and high incident energies. The effective interaction of the plasma charged particles is modelled by a pseudopotential which is a function of two parameters, namely the plasma screening strength and the de Broglie wavelength. A detailed study is made to explore the changes in the CSs of the above‐mentioned process with respect to the variation in the plasma screening strength and de Broglie wavelength. Significant changes are found to take place, when the screening strength and the de Broglie wavelength are varied. Specifically, the sharp minimum in the differential CS moves toward the forward direction with increasing de Broglie wavelength at a given screening strength.

Generalize for Future: Slow and Fast Trajectory Learning for CTR Prediction

Deep neural networks (DNNs) have achieved significant advancements in click-through rate (CTR) prediction by demonstrating strong generalization on training data. However, in real-world scenarios, the assumption of independent and identically distributed (i.i.d.) conditions, which is fundamental to this problem, is often violated due to temporal distribution shifts. This violation can lead to suboptimal model performance when optimizing empirical risk without access to future data, resulting in overfitting on the training data and convergence to a single sharp minimum. To address this challenge, we propose a novel model updating framework called Slow and Fast Trajectory Learning (SFTL) network. SFTL aims to mitigate the discrepancy between past and future domains while quickly adapting to recent changes in small temporal drifts. This mechanism entails two interactions among three complementary learners: (i) the Working Learner, which updates model parameters using modern optimizers (e.g., Adam, Adagrad) and serves as the primary learner in the recommendation system, (ii) the Slow Learner, which is updated in each temporal domain by directly assigning the model weights of the working learner, and (iii) the Fast Learner, which is updated in each iteration by assigning exponentially moving average weights of the working learner. Additionally, we propose a novel rank-based trajectory loss to facilitate interaction between the working learner and trajectory learner, aiming to adapt to temporal drift and enhance performance in the current domain compared to the past. We provide theoretical understanding and conduct extensive experiments on real-world CTR prediction datasets to validate the effectiveness and efficiency of SFTL in terms of both convergence speed and model performance. The results demonstrate the superiority of SFTL over existing approaches.

Nonmonotone variable projection algorithms for matrix decomposition with missing data

This paper investigates algorithms for matrix factorization when some or many components are missing, a problem that arises frequently in computer vision and pattern recognition. We demonstrate that the Jacobian used in the damped Wiberg (DW) method is exactly the same as that of Kaufman’s simplified variable projection (VP) algorithm. Our analysis provides a novel perspective on the efficiency of VP algorithms by improving the strong convexity of the approximate function. To enhance numerical stability, we set a lower bound on the damping parameter instead of adding a null space like the DW algorithm. Another challenge of low-rank matrix decomposition with missing data is the existence of many sharp local minima, which are often distributed in narrow valleys of the landscape of objection functions. Falling into such minima tends to result in poor reconstruction results. To address this issue, we design a non-monotonic VP algorithm, which can facilitate the algorithm to escape from sharp minima and converge to flatter minima. Numerical experiments confirm the effectiveness and efficiency of the proposed nonmonotone VP algorithm.

Improving Differentiable Architecture Search via self-distillation

Differentiable Architecture Search (DARTS) is a simple yet efficient Neural Architecture Search (NAS) method. During the search stage, DARTS trains a supernet by jointly optimizing architecture parameters and network parameters. During the evaluation stage, DARTS discretizes the supernet to derive the optimal architecture based on architecture parameters. However, recent research has shown that during the training process, the supernet tends to converge towards sharp minima rather than flat minima. This is evidenced by the higher sharpness of the loss landscape of the supernet, which ultimately leads to a performance gap between the supernet and the optimal architecture. In this paper, we propose Self-Distillation Differentiable Neural Architecture Search (SD-DARTS) to alleviate the discretization gap. We utilize self-distillation to distill knowledge from previous steps of the supernet to guide its training in the current step, effectively reducing the sharpness of the supernet’s loss and bridging the performance gap between the supernet and the optimal architecture. Furthermore, we introduce the concept of voting teachers, where multiple previous supernets are selected as teachers, and their output probabilities are aggregated through voting to obtain the final teacher prediction. Experimental results on real datasets demonstrate the advantages of our novel self-distillation-based NAS method compared to state-of-the-art alternatives.

Microwave-induced intersubband scattering and magneto-oscillation phenomena in an inhomogeneous 2D electron gas on liquid helium

The theoretical description of microwave-induced intersubband scattering in a 2D electron gas exposed to a magnetic field on liquid helium with the inhomogeneous distribution of subband excitation energies is presented. We demonstrate that the alignment of the staircases of Landau levels in the ground and the first excited subbands differently affects the microwave resonant absorption and in-plane magnetoconductivity. In contrast with previously used models, the microwave excitation rate is shown to have a strong dependence on the magnetic field with sharp minima at the Landau level alignment points even in a heavily inhomogeneous 2D electron system. The inhomogeneous distribution of transverse-quantization eigenfrequencies changes the lineshape of the intersubband resonance and substantially broadens conductivity magneto-oscillations. These results explain unexpected experimental observations reported previously.

Monitoring the ferroelectric phase transition of barium titanate ceramics via impulse excitation

Impulse excitation is a suitable tool to determine the temperature dependence of Young’s modulus and to monitor the ferroelectric phase transition of unpoled barium titanate ceramics (tetragonal-to-cubic and vice versa). By measurement from room temperature to 200–300 °C and back, it is found that there is an elastic anomaly with a sharp minimum (at 120 °C during heating and 110 °C during cooling, with a completely reproducible hysteresis), below and above which Young’s modulus exhibits values of 40–50 GPa and up to 90 GPa, respectively, and damping changes abruptly from values of order 0.01 to less than 0.001.

Thermal transport of fractionalized antiferromagnetic and field-induced states in the Kitaev material Na2Co2TeO6

We report an in-plane thermal transport study of the honeycomb Kitaev material Na$_2$Co$_2$TeO$_6$ at subKelvin temperatures. In zero field, the $\kappa(T)$ displays a rather weak $T$-dependence but has a non-zero residual term $\kappa_0/T$, indicating strong phonon scattering by magnetic excitation and the possibility of itinerant spinon-like excitations coexisting with an antiferromagnetic order below 27 K. We propose the zero-field ground state is a novel fractionalized antiferromagnetic (AF*) state with both magnetic order and fractionalized excitations. With both the heat current and external field along the $a*$ (Co-Co bond) direction, the $\kappa_{a*}$ exhibits two sharp minima at 7.5 T and 10 T, and its value at 8.5 T is almost the same as the pure phononic transport for the high-field polarized state. This confirms the phase boundaries of the reported field-induced intermediate state and suggest its gapless continuum excitations possibly transport heat. No such intermediate phase was found in the $\kappa_a$ for the current and field along the $a$ (zigzag chain) direction. Finally, Na$_2$Co$_2$TeO$_6$ displays a strongly anisotropic magneto-thermal conductivity since the in-plane (out-of-plane) field strongly enhances (suppresses) the $\kappa_{a*}$ and $\kappa_a$.

Electron transition dynamics in high-order harmonic generation process from H2+

The high-order harmonic generation from [Formula: see text] has been theoretically investigated by numerically solving the two-dimensional time-dependent Schrödinger equation. A sharp minimum can be seen in the harmonic spectrum for larger internuclear distance. Moreover, the electron transition process between the ground state and the first excited state competes with the ionization process at larger internuclear distance, which is the main reason of the spectral minimum. By superposing the static electric field and adjusting the orientation angle [Formula: see text], it is further proved that the electron transition frequency between two lowest electronic states can be decoded by the minimum.

Fuzzy hyperspheres via confining potentials and energy cutoffs

We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres , for all dimensions , initiated in Fiore and Pisacane (2018 J. Geom. Phys. 132 423–51). This is based on imposing a suitable energy cutoff on a quantum particle in subject to a confining potential well V(r) with a very sharp minimum on the sphere of radius r = 1; the cutoff and the depth of the well diverge with . As a result, the noncommutative Cartesian coordinates generate the whole algebra of observables on the Hilbert space ; applying polynomials in the to any we recover the whole . The commutators of the are proportional to the angular momentum components, as in Snyder noncommutative spaces. , as carrier space of a reducible representation of O(D), is isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) , which carries an irreducible representation of . Moreover, is isomorphic to . We resp. interpret , as fuzzy deformations of the space of (square integrable) functions on S d and of the associated algebra of observables, because they resp. go to as Λ diverges (with fixed). With suitable , in the same limit goes to the (algebra of functions on the) Poisson manifold ; more formally, yields a fuzzy quantization of a coadjoint orbit of that goes to the classical phase space .

Potential of TROPOMI for understanding spatio-temporal variations in surface NO2 and their dependencies upon land use over the Iberian Peninsula

Abstract. In orbit since late 2017, the Tropospheric Monitoring Instrument (TROPOMI) is offering new outstanding opportunities for better understanding the emission and fate of nitrogen dioxide (NO2) pollution in the troposphere. In this study, we provide a comprehensive analysis of the spatio-temporal variability of TROPOMI NO2 tropospheric columns (TrC-NO2) over the Iberian Peninsula during 2018–2021, considering the recently developed Product Algorithm Laboratory (PAL) product. We complement our analysis with estimates of NOx anthropogenic and natural soil emissions. Closely related to cloud cover, the data availability of TROPOMI observations ranges from 30 %–45 % during April and November to 70 %–80 % during summertime, with strong variations between northern and southern Spain. Strongest TrC-NO2 hotspots are located over Madrid and Barcelona, while TrC-NO2 enhancements are also observed along international maritime routes close the strait of Gibraltar, and to a lesser extent along specific major highways. TROPOMI TrC-NO2 appear reasonably well correlated with collocated surface NO2 mixing ratios, with correlations around 0.7–0.8 depending on the averaging time. We investigate the changes of weekly and monthly variability of TROPOMI TrC-NO2 depending on the urban cover fraction. Weekly profiles show a reduction of TrC-NO2 during the weekend ranging from −10 % to −40 % from least to most urbanized areas, in reasonable agreement with surface NO2. In the largest agglomerations like Madrid or Barcelona, this weekend effect peaks not in the city center but in specific suburban areas/cities, suggesting a larger relative contribution of commuting to total NOx anthropogenic emissions. The TROPOMI TrC-NO2 monthly variability also strongly varies with the level of urbanization, with monthly differences relative to annual mean ranging from −40 % in summer to +60 % in winter in the most urbanized areas, and from −10 % to +20 % in the least urbanized areas. When focusing on agricultural areas, TROPOMI observations depict an enhancement in June–July that could come from natural soil NO emissions. Some specific analysis of surface NO2 observations in Madrid show that the relatively sharp NO2 minimum used to occur in August (drop of road transport during holidays) has now evolved into a much broader minimum partly de-coupled from the observed local road traffic counting; this change started in 2018, thus before the COVID-19 outbreak. Over 2019–2021, a reasonable consistency of the inter-annual variability of NO2 is also found between both datasets. Our study illustrates the strong potential of TROPOMI TrC-NO2 observations for complementing the existing surface NO2 monitoring stations, especially in the poorly covered rural and maritime areas where NOx can play a key role, notably for the production of tropospheric O3.

Effect of Uniform Strain on Graphene Surface Plasmon Excitations

This paper reports a study of the reflectance of the optical sensors based on graphene under uniform strain. Assuming the graphene layer is surrounded by two different semi-infinite dielectric media, the generalized Fresnel coefficients are derived as a function of usual quantities (e.g., dielectric constants, incident angles, and strain) and anisotropic optical conductivity. The strain not only changes the electronic band structure but also can be employed to tune the electronic collective excitations (plasmons) and thus the optical reflectance of graphene monolayers. One of the most common techniques for plasmon excitation is the Kretschmann configuration. It is based on the observation of a sharp minimum in the reflection coefficient versus the angle (or wavelength) curve. Because strain induces anisotropy in graphene optical conductivity the strain-dependent orientation plays an important role to manipulate the variation of graphene plasmon energy, which may be useful to synchronize graphene properties in plasmonic devices to enhance light-matter interactions.

Nonlinear Relaxation Time in Glassy Poly(methyl methacrylate) Subjected to Tensile Creep Stresses

Nonlinear viscoelastic behavior of glassy poly(methyl methacrylate) (PMMA) subjected to high tensile creep stresses was studied. Right after the loading of creep stress, a rapid decrease of strain rate as a function of strain was observed, indicating behavior of linear viscoelastic solid. Then after showing a sharp minimum, the strain rate gradually increased due to nonlinear flow and reached to almost constant maximum values at steady flow state, and then slightly decreased. Strain range in which the nonlinear flow was observed did not strongly depend on the magnitude of creep stress and temperature, although magnitude of strain rate was obviously affected by these conditions. Nonlinear single relaxation time τ for the glass in the nonlinear flow range of strain was evaluated by means of a simple mechanical model. The nonlinear relaxation time τ decreased with increasing time t to a steady value appeared in the steady flow state. A quantity τ/t was found to be an almost unique decreasing function of strain independently of differences in magnitude of creep stress and temperature. This characteristic dependence of relaxation time τ on strain was considered as a result of a constant decreasing slope of τ-t relation, which slightly depends on the magnitude of stress, during transient flow state under constant high stresses.

Escape dynamics of a particle from a purely nonlinear truncated quartic potential well under harmonic excitation

This paper focuses on the escape problem of a harmonically forced classical particle from a purely quartic truncated potential well. The latter corresponds to various engineering systems that involve purely cubic restoring force and absence of linear stiffness even under the assumption of small oscillations, such as pre-tensioned metal wires and springs, and compliant structural components made of polymer materials. This is in contrast to previous studies where the equivalent potential well could be treated as linear at first approximation under the assumption of small perturbations. Due to the strong nonlinearity of the current potential well, traditional analytical methods are inapplicable for describing the transient bounded and escape dynamics of the particle. The latter is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle variables and the corresponding reduced resonance manifold. The escape envelope is formulated analytically. Surprisingly, despite the essential nonlinearity of the well investigated, it exhibits a universal property of a sharp minimum due to the existence of multiple intersecting escape mechanisms. Unlike previous studies, three underlying mechanisms that govern the transient dynamics of the particle were identified: two maximum mechanisms and a saddle mechanism. The first two correspond to a gradual increase in the system’s response amplitude for a proportional increase in the excitation intensity, and the latter corresponds to an abrupt increase in the system’s response and therefore more potentially hazardous. The response of the particle is described in terms of energy-based response curves. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. All theoretical predictions are in complete agreement with numerical results.

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ РАВНОВЕСИЯ СТОЛБА СЖИМАЕМОЙ АТМОСФЕРЫ. ЧАСТЬ 1: СТАЦИОНАРНЫЕ РЕШЕНИЯ ДЛЯ ТЕМПЕРАТУРЫ

An analytical model of mathematical physics, which is implemented for an unbounded column of compressible atmosphere, is discussed. For the column, there are two equilibrium conditions, which are written in the form of a system of aerodynamic equations. The first condition is the classical condition of hydrostatic, which is derived from the equations of motion. The second one is derived from the continuity equation and arises due to temporal variations of air density. Two boundary conditions are selected near the solid surface: one is the magnitude of temperature and second condition is the magnitude of its vertical gradient. This allows us to consider the column without any restrictive conditions at some upper boundary. The analytical solution for adiabatic temperature can contain a sharp minimum at some height, above which the temperature rises with altitude. Herewith the temperature profile depends on the altitude linearly near the surface. The physical interpretation of the solutions based on the data of experimental measurements in the Earth's atmosphere is discussed. The analytical solution can be used to describe adiabatic atmospheric column, which is important for some applications. The research results improve understanding of processes in atmosphere and can be used for scientific and educational purposes.

Learning through atypical phase transitions in overparameterized neural networks.

Current deep neural networks are highly overparameterized (up to billions of connection weights) and nonlinear. Yet they can fit data almost perfectly through variants of gradient descent algorithms and achieve unexpected levels of prediction accuracy without overfitting. These are formidable results that defy predictions of statistical learning and pose conceptual challenges for nonconvex optimization. In this paper, we use methods from statistical physics of disordered systems to analytically study the computational fallout of overparameterization in nonconvex binary neural network models, trained on data generated from a structurally simpler but "hidden" network. As the number of connection weights increases, we follow the changes of the geometrical structure of different minima of the error loss function and relate them to learning and generalization performance. A first transition happens at the so-called interpolation point, when solutions begin to exist (perfect fitting becomes possible). This transition reflects the properties of typical solutions, which however are in sharp minima and hard to sample. After a gap, a second transition occurs, with the discontinuous appearance of a different kind of "atypical" structures: wide regions of the weight space that are particularly solution dense and have good generalization properties. The two kinds of solutions coexist, with the typical ones being exponentially more numerous, but empirically we find that efficient algorithms sample the atypical, rare ones. This suggests that the atypical phase transition is the relevant one for learning. The results of numerical tests with realistic networks on observables suggested by the theory are consistent with this scenario.

New Polarimetric Data for the Galilean Satellites: Europa Observations and Modeling

This paper is dedicated to a long-standing problem of the shape of the negative branch of polarization (NBP) for Jupiter's moon Europa, determination of which is crucial for the characterization of the icy regolith on this satellite and similar objects, as well as for further progress in understanding light scattering by particulate surfaces. To establish the shape of Europa's NBP, in 2018–2021 we accomplished high-precision disk-integrated polarimetry of Europa in the UBVR I bands using the identical two-channel photoelectric polarimeters mounted on the 2.6 m Shajn reflector of the Crimean Astrophysical Observatory and the 2 m telescope of the Peak Terskol Observatory. We found that the polarization dependence on the phase angle in each filter is an asymmetric curve with a sharp polarization minimum at phase angle , after which the polarization degree gradually increases to positive values, passing the inversion angle at α inv ≈ 6° − 7°. Within the error limits, the parameters P min, , and α inv of the NPB are independent of the wavelength in the visible spectrum. The polarization curve clearly demonstrates the so-called polarization opposition effect (POE). Our analysis of the previous and new polarimetric observations of Europa allows us to conclude that the POE is caused by coherent backscattering of sunlight on microscopic icy grains covering Europa’s surface. Computer modeling with the numerical radiative transfer coherent backscattering method demonstrates that the best fit to the polarimetric observations and geometric albedo of Europa is provided by a regolith layer of elementary single-scattering albedo ∼0.985 and extinction mean free path length 2π l/λ eff ≈ 150, λ eff representing the effective wavelength in the UBVR I spectral bands.

Субградиентные методы для задач негладкой оптимизации с некоторой релаксацией условия острого минимума

In this paper we propose a generalized condition for a sharp minimum, somewhat similar to the inexact oracle proposed recently by Devolder-Glineur-Nesterov. The proposed approach makes it possible to extend the class of applicability of subgradient methods with the Polyak step-size, to the situation of inexact information about the value of the minimum, as well as the unknown Lipschitz constant of the objective function. Moreover, the use of local analogs of the global characteristics of the objective function makes it possible to apply the results of this type to wider classes of problems. We show the possibility of applying the proposed approach to strongly convex non-smooth problems, also, we make an experimental comparison with the known optimal subgradient method for such a class of problems. Moreover, there were obtained some results connected to the applicability of the proposed technique to some types of problems with convexity relaxations: the recently proposed notion of weak $\beta$-quasi-convexity and ordinary quasi-convexity. Also in the paper, we study a generalization of the described technique to the situation with the assumption that the $\delta$-subgradient of the objective function is available instead of the usual subgradient. For one of the considered methods, conditions are found under which, in practice, it is possible to escape the projection of the considered iterative sequence onto the feasible set of the problem.

Experimentally evaluating plasmonic sensing performance of silver film over nanosphere surface (AgFON)

In the present work, Silver film over nanosphere surface (AgFON) structures were fabricated on a self-assembled monolayer (SAM) of polystyrene spheres (200 nm) by a simple and cost effective drop-casting followed by thermal evaporation techniques. The thickness of Ag thin film was varied from 20 nm to 100 nm in a step of 20 nm. Field emission scanning electron microscopy (FESEM) revealed that the morphology of AgFON changes from nano island to nanoshells with increasing thickness. Reflection spectra of AgFON of thickness >60 nm exhibited a sharp minimum due to the excitation of cavity mode plasmon. Plasmonic sensing capabilities of AgFON have been investigated with respect its thickness. AgFON of 100 nm exhibited a bulk sensitivity of 632.54 nm/RIU while 80 nm showed a sensitivity of 365 nm/RIU towards a thin layer of volatile organic compounds such as ethanol, toluene and isopropyl alcohol. Biological molecules such as urea, creatinine, glucose, melamine and glutathione have been tested with the AgFON. The AgFON displays a good capability of detecting 1 mM creatinine in an aqueous solution. A successful attempt has been made to detect the creatinine of >1 mM in human urine.